ExampleConstruction: Literature-sourcedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-11

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The Cantor function is continuous and of bounded variation but not absolutely continuous

Example

The Cantor function $c:[0,1]\to[0,1]$ is continuous and nondecreasing, so it has total variation one, but it is not absolutely continuous.

Facts & Assumptions

Given: The Cantor function and its standard stage construction.

[L1]

The Cantor function is continuous and nondecreasing, with $c(0)=0$ and $c(1)=1$ (The Cantor function is continuous on $[0,1]$ , The Cantor function is well defined, satisfies $c(x) \le c(y)$ whenever $x \le y$ , is surjective onto $[0,1]$ , and is constant on every interval removed from the Cantor set).

[L2]

At stage $n$ , the $2^n$ surviving closed intervals have total length $(2/3)^n$ , and $c$ increases by $2^{-n}$ across each.

[L3]

The sequence $(2/3)^n$ tends to zero (For $|r| < 1$ the sequence $r^k$ is null, and for $|r| > 1$ the sequence $|r|^k$ diverges to $+\infty$ ).

Verification

technique · direct

1.1

Monotonicity makes every variation sum telescope after its absolute values are removed, so $\operatorname{Var}_{[0,1]}(c)=c(1)-c(0)=1$ . Thus $c$ is BV.

2.1

For the finite disjoint family of the $2^n$ surviving intervals, the total length is $(2/3)^n$ but the sum of endpoint increments is $2^n2^{-n}=1$ . By [L3], the former is eventually smaller than every prescribed $\delta>0$ , while the latter never falls below, say, $1/2$ . This contradicts the definition of absolute continuity.

L2L3∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 113 results over 22 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

Christopher Heil, Absolute Continuity and the Banach-Zaretsky Theorem, Example 3.3 (standard reference, not scraped)